Generally we can think of conjugacy as when we know the distribution of a parameter in another distribution.
pcvr::conjugate
In pcvr 11 distributions are supported in the conjugate function.
We’ll go over those distributions, what they tend to represent, how they are updated, and what the common alternative tests would be for that kind of data.
The rope_range and rope_ci arguments allow region-of-practical-equivalence (ROPE) testing using the difference in the posterior distributions if two samples are given.
The support argument optionally lets you set the support range for the distribution. In practice this exists for internal use and is not something you would generally want to set.
Here we’d simply conclude that there is about a 19% chance that the germination rate is the same between these two genotypes after 1 week. We could do more with ROPE testing, but we’ll do that in the next example.
Area
Lots of phenotypes are gaussian and conjugate can be used similarly to a T-test with the “t” method. Consider area data that looks like this example.
Area
Here we include a ROPE test corresponding to our belief that any difference in Area of \(\pm2 cm^2\) is biologically insignificant. We also show the formula syntax and use non-default priors here (since default priors include negative values which can’t happen with area).
res <-conjugate(s1 = y ~ geno, s2 = df,method ="t",plot =TRUE,rope_range =c(-2, 2),priors =list(mu =10, n =1, s2 =3),hypothesis ="unequal")
Area
Our plot shows about a 83% chance that these distributions are unequal and a 24% chance that the difference in means is within \(\pm2 cm^2\).
Area
The other aspects of the output are a summary and the posterior parameters.
The posterior is the prior list updated with the given data, this allows for Bayesian updating hypothetically.
do.call(rbind, res$posterior)
mu n s2
[1,] 15.00699 11 9.383789
[2,] 11.93384 11 16.24362
Leaf Counts
There are also several phenotypes that are counts. Numbers of vertices, leaves, flowers, etc could all be used with one of the count distributions. Here we consider Poisson distributed leaf counts between two genotypes.
Leaf Counts
Here we model \(X \sim Poisson(\lambda)\\ \lambda \sim \Gamma(A, B)\)
res <-conjugate(s1 = y ~ geno, s2 = df,method ="poisson",plot =TRUE,rope_range =c(-1, 1),priors =list(a =1, b =1),hypothesis ="unequal")
Leaf Counts
We can comfortably say that the difference in the posteriors is not in [-1, 1] and there is a 91% chance that the Gamma distributions for \(\lambda\) are different.
Hue
Finally, we’ll show an example using what is likely the least-familiar distribution in conjugate, the Von-Mises distribution.
The Von-Mises distribution is a symmetric circular distribution defined on \([-\pi, \pi]\).
To use Von-Mises with data on other intervals there is a boundary element in the prior that is used to rescale data to radians for the updating before rescaling back to the parameter space. See ?conjugate for more examples of the boundary.
Hue
Note this is a very exaggerated example for the plant phenotyping setting since green happens to be in the middle of the hue circle, which wraps in the red colors.
If you do have wrapped circular data then looking at it in a non-circular space like this would be a problem. For values we normally get from plants other continuous methods can generally be a decent approximation.
Hue
res <-conjugate(s1 = y ~ geno, s2 = df,method ="vonmises2",priors =list(mu =0, kappa =0.5, boundary =c(-pi, pi), n =1),plot =TRUE)
Hue
Here our distributions are very similar and there is about a 79% chance that their parameters are the same given this data and our (very wide) priors.
Note that if you use data in a different boundary space than radians the rope_range would be given in the observed parameter space, not in radians.
Conclusion
Hopefully this has been a useful introduction to Bayesian conjugacy.
Please direct questions/issues with pcvr::conjugate to the pcvr github issues or help-datascience slack channel for DDPSC users.
